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For a given set of data randomly sampled from a Normal distribution, whose
mean m
is unknown and unknown standard deviation s,
the distribution of uncertainty of the true standard deviation is calculated
from the formula:
where 
is a chi-squared
distribution with n-1 degrees of freedom. It has a mean of (n-1)
so the term in the square root is centred on 1, which means that the uncertainty
distribution for s is centred
around 
.
[This page provides an explanation of the derivation]
This 
spreadsheet example model
lets you generate values for the above uncertainty distribution for s
for a data set.